Bruno Drieux and Gabriel Merlin, Ph.D. students and scholars of the Grothendieck Institute, participated in the international conference TACL 2026 (Topology, Algebra, and Categories in Logic), held in Kraków, Poland, from 27 to 31 July 2026.
Both presented joint work with Olivia Caramello contributing to the development of relative topos theory and its categorical foundations.
Bruno Drieux’s talk, “Relative (co-)Yoneda Lemma for Categories Indexed over a Small-Generated Site“, presented relative versions of the Yoneda and co-Yoneda lemmas, fundamental results describing how mathematical objects can be understood through their relationships with other objects.
His presentation also developed tools for indexed limits and colimits, providing foundations for category theory over an arbitrary base topos.
Gabriel Merlin’s talk, “Relative Points and Completions of Indexed Categories“, examined how classical categorical constructions can be extended to a relative setting, in which mathematical structures are studied over a base topos rather than over sets.
In particular, he presented geometric descriptions of indexed pro-completions and indexed Karoubi envelopes in terms of relative points of toposes, illustrating the theory with an application to Boolean algebras equipped with group actions.
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